Generalised hypergeometric function $_4F_3$ at unit argument

Asked Jun 15 '20 at 15:17

Active Jun 15 '20 at 15:17

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Is there a way to simplify the following generalised hypergeometric function evaluated at unit argument

$$ \, _4F_3\left(-n,3+n,\frac{2}{3},\frac{5}{3};2,-\frac{1}{3}-n,\frac{8}{3}+n;1\right) $$

with $n\geq0$ being an integer?

asked Jun 15 '20 at 15:17

user12588

Might be worth trying Dougall's formula here: https://en.wikipedia.org/wiki/Generalized_hypergeometric_function
That covers quite a few of these kind of problems. Will want three of the parameters to be the same value so they cancel out.
– Benedict W. J. Irwin Jun 15 '20 at 15:21
I thought Dougall's formula was only valid for the $_7F_6$... – user12588 Jun 15 '20 at 16:54
That's why you need to cancel some of the terms out, i.e. $$ ;_7 F_6(a,b,c,d,1,1,1;e,f,g,1,1,1;1) = ;_4 F_3(a,b,c,d;e,f,g;1) $$ although it might not be as easy as that. Just an idea. – Benedict W. J. Irwin Jun 15 '20 at 17:00
That is a simple idea that I did not think about. Thanks, will try. – user12588 Jun 15 '20 at 17:12
Alas, it does not seem to work. Thanks for the tip anyway. – user12588 Jun 15 '20 at 17:26

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